The steam client don't have to be x86_64 bit in order to run x86_64 bit game.Example :

I have ut2004 in my library which runs as x86_64 bit and the steam overlay works fine.

There separate processes.

With regards the "x64 " misunderstanding, Blame windows lingo. Seem to remember "x64" denoting sparc proc's though way back in the day which is why it's writen x86_64. But not sure on that, My memory ain't what it used to be. ;)